Bivariate C1C1 cubic spline spaces with homogeneous boundary conditions over FVS triangulation

In this paper, we mainly generalize the results in [H.W. Liu, D. Hong, D.Q. Cao, Bivariate C1C1 cubic spline space over a nonuniform type-2 triangulation and its subspaces with boundary conditions, Comput. Math. Appl. 49 (2005), 1853–1865] from the type-2 triangulation to the so-called FVS triangulation (a triangulated quadrangulation). We study the bivariate C1C1 cubic spline spaces S31,0(♦̃) and S31,1(♦̃) with homogeneous boundary conditions over an FVS triangulation ♦̃. The dimensions are obtained and the locally supported bases are constructed for these spline spaces. Furthermore, we also study the explicit Bézier ordinates of the interpolation basis splines on a representative triangulated quadrilateral. The results of this paper can be applied in many fields such as the finite element method for partial differential equation, computer aided geometric design, numerical approximation, and so on.